Laura owes $18,000 on her student loans. The interest rate on the bank loan is 2.5% and the interest rate on the federal loan is 6.9%. The total amount of interest she paid last year was $1,066. What was the principal for each loan?

The principal of the bank loan is $4,000

The principal of the federal lone is $14,000

Step-by-step explanation:

* Lets explain how to solve the problem

- Laura owes $18,000 on her student loans

- The interest rate on the bank loan is 2.5%

- The interest rate on the federal loan is 6.9%

- The total amount of interest she paid last year was $1,066

* Assume that the principle of the bank loan is $x and The principal of

the federal loan is $y

∵ Her loan is 18,000

∴ x + y = 18,000 ⇒ (1)

- The interest is I = PRT, where P is the principal, R is the rate of interest

in decimal ant is the time

# The bank lone

∵ P = $x

∵ R = 2.5/100 = 0.025

∵ T = 1

∴ The interest of the bank lone = x (0.025) (1) = 0.025 x

# The federal lone

∵ P = $y

∵ R = 6.9/100 = 0.069

∵ T = 1

∴ The interest of the federal lone = 9 (0.069) (1) = 0.069 y

- The total interest is the sum of the interest of the bank lone and the

interest of the federal loan

∵ The total interest is $1,066

∴ 0.025 x + 0.069 y = 1,066 ⇒ (2)

* We have system of equations

- Multiply equation (1) by - 0.025 to eliminate x

∴ - 0.025 x - 0.025 y = - 450 ⇒ (3)

- Add equations (2) and (3)

∴ 0.044 y = 616

- Divide both sides by 0.044

∴ y = 14,000

- Substitute value of y in equation (1) to find x

∴ x + 14000 = 18000

- Subtract 18000 from both sides

∴ x = 4000

∵ x represents the principal of the bake loan and y represents the

principal of the federal lone

* The principal of the bank loan is $4,000

* The principal of the federal lone is $14,000